Bottom Line:
We have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network.We showed that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length.We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.

ABSTRACTWe have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network. We showed that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. In this paper, we explore how frequency-encoding of axonal length can be decoded by a frequency-modulated gene network. If the protein output were thresholded, then this could provide a mechanism for axonal length control. We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.

Figure 2: Frequency of periodic solutions plotted against axonal length. [Plot was obtained by looking at the power spectrum of the retrograde signal and taking the frequency of the signal to be where the sharp peak of the spectrum occurred.] Insets show time series plots at specific values of the delay generated using the dde23 program in MATLAB: (A) τ = 1, (B) τ = 2, (C) τ = 10. Other parameter values are n = 4, I0 = 10, WE = WI = 9.5 such that τc ≈ 1.5.

Mentions:
where ω denotes the frequency of the periodic solution and for P = E, I and the steady state of the corresponding chemical signal. We immediately notice two facts from Equation (4). First, if τ = 0, then the bifurcation conditions cannot be satisfied, suggesting that there exists a critical delay τc past which Equations (1) and (2) have periodic solutions. This corresponds with the existence of some critical axonal length Lc past which signals will oscillate. Second, the bifurcation conditions can only be satisfied if . It follows that the feedback strengths WP must be sufficiently large and/or the Hill function must be sufficiently steep. The latter implies that oscillations are facilitated if the chemical signal interactions are cooperative in nature, as reflected by the value of n in the delayed feedback model. The existence of the Hopf bifurcation point does not in itself guarantee the onset of stable limit cycles for τ > τc. However, this can be verified numerically, and one finds that the frequency of the oscillation beyond the bifurcation point is a monotonically decreasing function of L, see Figure 2.

Figure 2: Frequency of periodic solutions plotted against axonal length. [Plot was obtained by looking at the power spectrum of the retrograde signal and taking the frequency of the signal to be where the sharp peak of the spectrum occurred.] Insets show time series plots at specific values of the delay generated using the dde23 program in MATLAB: (A) τ = 1, (B) τ = 2, (C) τ = 10. Other parameter values are n = 4, I0 = 10, WE = WI = 9.5 such that τc ≈ 1.5.

Mentions:
where ω denotes the frequency of the periodic solution and for P = E, I and the steady state of the corresponding chemical signal. We immediately notice two facts from Equation (4). First, if τ = 0, then the bifurcation conditions cannot be satisfied, suggesting that there exists a critical delay τc past which Equations (1) and (2) have periodic solutions. This corresponds with the existence of some critical axonal length Lc past which signals will oscillate. Second, the bifurcation conditions can only be satisfied if . It follows that the feedback strengths WP must be sufficiently large and/or the Hill function must be sufficiently steep. The latter implies that oscillations are facilitated if the chemical signal interactions are cooperative in nature, as reflected by the value of n in the delayed feedback model. The existence of the Hopf bifurcation point does not in itself guarantee the onset of stable limit cycles for τ > τc. However, this can be verified numerically, and one finds that the frequency of the oscillation beyond the bifurcation point is a monotonically decreasing function of L, see Figure 2.

Bottom Line:
We have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network.We showed that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length.We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.

ABSTRACTWe have recently developed a mathematical model of axonal length sensing in which a system of delay differential equations describe a chemical signaling network. We showed that chemical oscillations emerge due to delayed negative feedback via a Hopf bifurcation, resulting in a frequency that is a monotonically decreasing function of axonal length. In this paper, we explore how frequency-encoding of axonal length can be decoded by a frequency-modulated gene network. If the protein output were thresholded, then this could provide a mechanism for axonal length control. We analyze the robustness of such a mechanism in the presence of intrinsic noise due to finite copy numbers within the gene network.